Scaling down of data

ABSTRACT

The scaling down of data is provided. At least two blocks of transformed data samples representing at least two blocks of original data samples are received. One of at least two tables of constants is selected wherein each table of constants is capable of reducing the number of transformed data samples by a different factor. The constants taken from the selected table are applied to the at least two blocks of transformed data samples to produce one block of transformed data samples representing one block of final data samples. The data is processed one dimension at a time by multiplying the data in one dimension with selected constants taken from previously developed tables corresponding to the desired scale down factor. Scaling down by different factors in each dimension as well as scaling down in one dimension and scaling up in the other dimension may be achieved. In addition, the de-quantization of the quantized transform coefficients may be accomplished by pre-multiplication of the selected constants when the quantization values are known. In a similar way the re-quantization may be accomplished by a pre-division of the selected constants. Both de-quantization and re-quantization may be combined when the input quantized transform coefficients and output quantized transform coefficients are desired.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application is related to the following U.S. patentapplications: U.S. Pat. No. 6,256,422 issued Jul. 3, 2001, with Ser. No.09/186,245 filed Nov. 4, 1998, by Joan L. Mitchell and Martin J. Brightfor “Transform-Domain Correction of Real Domain Errors” (IBM DocketYO998-372); U.S. Pat. No. 6,393,155 issued May 21, 2002, with Ser. No.09/186,249 filed Nov. 4, 1998 by Martin J. Bright and Joan L. Mitchellfor “Error Reduction in Transformed Digital Data” (IBM DocketYO998-373); U.S. Pat. No. 6,535,645 issued Mar. 18, 2003 and which is adivisional application of U.S. Pat. No. 6,393,155; Ser. No. 09/186,247filed Nov. 4, 1998 by Martin J. Bright and Joan L. Mitchell for“Reduced-error Processing of Transformed Digital Data” (IBM DocketYO998-331); U.S. Pat. No. 6,671,414 issued Dec. 30, 2003, with Ser. No.09/524,266, filed Mar. 13, 2003, by Charles A. Micchelli, Marco Martens,Timothy J. Trenary and Joan L. Mitchell for “Shift and/or Merge ofTransformed Data Along One Axis” (IBM Docket YO999-313); U.S. Pat. No.6,678,423 issued Jan. 13, 2004, with Ser. No. 09/524,389, filed Mar. 13,2003, by Timothy J. Trenary, Joan L. Mitchell, Charles A. Micchelli, andMarco Martens for “Shift and/or Merge of Transformed Data Along TwoAxes” (IBM Docket YO999-337); and Ser. No. 09/570,382, filed May 12,2000, by Timothy J. Trenary, Joan L. Mitchell, Ian R. Finlay and NenadRijavec for “Method and Apparatus For The Scaling Up Of Data” (IBMDocket BLD9-99-0041), all assigned to a common assignee with thisapplication and the disclosures of which are incorporated herein byreference.

This application is a continuation of and claims the benefit ofapplication Ser. No. 10/983,080, filed on Nov. 4, 2004, which is adivisional of and claims the benefit of application Ser. No. 09/570,849filed on May 12, 2000, the disclosure of each of which is incorporatedherein by reference in its entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to a method and apparatus for efficiently scalingdown data which has been transformed from the real domain.

2. Description of the Related Art

Many types of data, such as radar data, oil well log data and digitalimage data, can consume a large amount of computer storage space. Forexample, computerized digital image files can require in excess of 1 MB.Therefore, several formats have been developed which manipulate the datain order to compress it. The discrete cosine transform (DCT) is a knowntechnique for data compression and underlies a number of compressionstandards.

The mathematical function for a DCT in one dimension is:

${{\overset{\sim}{s}(k)} = {{c(k)}{\sum\limits_{n = 0}^{N - 1}{{s(n)}\cos \frac{{\pi \left( {{2n} + 1} \right)}k}{2N}}}}},$

where s is the array of N original values, {tilde over (s)} is the arrayof N transformed values and the coefficients c are given by

${{c(k)} = {{\sqrt{\frac{1}{N}}\mspace{14mu} {for}\mspace{14mu} k} = 0}};{{c(k)} = {{\sqrt{\frac{2}{N}}\mspace{14mu} {for}\mspace{14mu} k} > 0}}$

Taking for example the manipulation of image data, blocks of dataconsisting of 8 rows by 8 columns of data samples frequently areoperated upon during image resizing processes. Therefore atwo-dimensional DCT calculation is necessary. The equation for atwo-dimensional DCT where N=8 is:

${{\overset{\sim}{s}\left( {i,j} \right)} = {{c\left( {i,j} \right)}{\sum\limits_{n = 0}^{7}{\sum\limits_{m = 0}^{7}{{s\left( {m,n} \right)}\cos \frac{{\pi \left( {{2m} + 1} \right)}i}{16}\cos \frac{{\pi \left( {{2n} + 1} \right)}j}{16}}}}}},$

where s is an 8×8 matrix of 64 values; {tilde over (s)} is an 8×8 matrixof 64 coefficients and the constants c(i,j) are given by

${{c\left( {i,j} \right)} = \frac{1}{8}},{{{{when}\mspace{14mu} i} = {{0\mspace{14mu} {and}\mspace{14mu} j} = 0}};}$${{c\left( {i,j} \right)} = {{\frac{1}{4\sqrt{2}}\mspace{14mu} {if}\mspace{14mu} i} = {{{0\mspace{14mu} {and}{\; \mspace{11mu}}j} > {0\mspace{14mu} {or}{\mspace{11mu} \;}i} > {0\mspace{14mu} {and}\mspace{14mu} j}} = 0}}};$${{{c\left( {i,j} \right)} = {\frac{1}{4}\mspace{14mu} {when}\mspace{14mu} i}},{j > 0}}\;$

Because data is taken from the “real” or spatial image domain andtransformed into the DCT domain by equations (1) and (2), these DCToperations are referred to as forward Discrete Cosine Transforms (FDCT),or forward transform operations.

As previously mentioned, the DCT is an image compression technique whichunderlies a number of compression standards. These include thewell-known Joint Photographic Experts Group (JPEG) and the MovingPicture Experts Group (MPEG) standards. Comprehensive references on theJPEG and MPEG standards include JPEG Still Image Data CompressionStandard by William B. Pennebaker and Joan L. Mitchell (© 1993 VanNostrand Reinhold), and MPEG Video Compression Standard by Joan L.Mitchell, William B. Pennebaker, et al (© 1997 Chapman & Hall).

Looking at the JPEG method, for example, there are five basic steps.Again taking the example of the manipulation of image data, the firststep is to extract an 8×8 pixel block from the image. The second step isto calculate the FDCT for each block. Third, a quantizer rounds off theDCT coefficients according to the specified image quality. Fourth, thequantized, two-dimensional 8×8 block of DCT coefficients are reorderedinto a one-dimensional vector according to a zig zag scan order. Fifth,the coefficients are compressed using an entropy encoding scheme such asHuffman coding or arithmetic coding. The final compressed data is thenwritten to the output file.

Returning to the first step, source image samples are grouped into 8×8data matrices, or blocks. The initial image data is frequently convertedfrom normal RGB color space to a luminance/chrominance color space, suchas YUV. YUV is a color space scheme that stores information about animage's luminance (brightness) and chrominance (hue). Because the humaneye is more sensitive to luminance than chrominance, more informationabout an image's chrominance can be discarded as compared to luminancedata.

Once an 8×8 data block has been extracted from the original image and isin the desired color scheme, the DCT coefficients are computed. The 8×8matrix is entered into the DCT algorithm, and transformed into 64unique, two-dimensional spatial frequencies thereby determining theinput block's spectrum.

The ultimate goal of this FDCT step is to represent the image data in adifferent domain using the cosine functions. This can be advantageousbecause it is a characteristic of cosine functions that most of thespatial frequencies will disappear for images in which the image datachanges slightly as a function of space. The image blocks aretransformed into numerous curves of different frequencies. Later, whenthese curves are put back together through an inverse step, a closeapproximation to the original block is restored.

After the FDCT step, the 8×8 matrix contains transformed data comprisedof 64 DCT coefficients in which the first coefficient, commonly referredto as the DC coefficient, is related to the average of the original 64values in the block. The other coefficients are commonly referred to asAC coefficients.

Up to this point in the JPEG compression process, little actual imagecompression has occurred. The 8×8 pixel block has simply been convertedinto an 8×8 matrix of DCT coefficients. The third step involvespreparing the matrix for further compression by quantizing each elementin the matrix. The JPEG standard gives two exemplary tables ofquantization constants, one for luminance and one for chrominance. Theseconstants were derived from experiments on the human visual system. The64 values used in the quantization matrix are stored in the JPEGcompressed data as part of the header, making dequantization of thecoefficients possible. The encoder needs to use the same constants toquantize the DCT coefficients.

Each DCT coefficient is divided by its corresponding constant in thequantization table and rounded off to the nearest integer. The result ofquantizing the DCT coefficients is that smaller, unimportantcoefficients will disappear and larger coefficients will loseunnecessary precision. As a result of this quantization step, some ofthe original image quality is lost. However, the actual image data lostis often not visible to the human eye at normal magnification.

Quantizing produces a list of streamlined DCT coefficients that can nowbe very efficiently compressed using either a Huffman or arithmeticencoding scheme. Thus the final step in the JPEG compression algorithmis to encode the data using an entropy encoding scheme. Before thematrix is encoded, it is arranged in a one-dimensional vector in azigzag order. The coefficients representing low frequencies are moved tothe beginning of the vector and the coefficients representing higherfrequencies are placed towards the end of the vector. By placing thehigher frequencies (which are more likely to be zeros) at the end of thevector, an end of block code truncates the larger sequence of zeroswhich permits better overall compression.

Equations (1) and (2) describe the process for performing a FDCT, i.e.,taking the data from the real domain into the DCT domain. When it isnecessary to reverse this step, i.e., transform the data from the DCTdomain to the real domain, a DCT operation known as an Inverse DiscreteCosine Transform (IDCT), or an inverse transform operation, can beperformed. For a one-dimensional, inverse transform operation, the IDCTis defined as follows:

${{s(k)} = {\sum\limits_{n = 0}^{N - 1}{{c(k)}{\overset{\sim}{s}(k)}\cos \frac{{\pi \left( {{2n} + 1} \right)}k}{2N}}}},$

where s is the array of N original values, {tilde over (s)} is the arrayof N transformed values and the coefficients c are given by

${{c(k)} = {{\sqrt{\frac{1}{N}}\mspace{14mu} {for}\mspace{14mu} k} = 0}};{{c(k)} = {{\sqrt{\frac{2}{N}}\mspace{14mu} {for}\mspace{14mu} k} > 0}}$

For an inverse transform operation in two dimensions where N=8, the IDCTis defined:

$\begin{matrix}{{{s\left( {m,n} \right)} = {\sum\limits_{i = 0}^{7}{\sum\limits_{j = 0}^{7}{{c\left( {i,j} \right)}{\overset{\sim}{s}\left( {i,j} \right)}\cos \frac{{\pi \left( {{2m} + 1} \right)}i}{16}\cos \frac{{\pi \left( {{2n} + 1} \right)}j}{16}}}}},} & (4)\end{matrix}$

where s is an 8×8 matrix of 64 values, S is an 8×8 matrix of 64coefficients and the constants c(i,j) are given by

${{c\left( {i,j} \right)} = {{\frac{1}{8}\mspace{14mu} {when}\mspace{14mu} i} = {{0\mspace{14mu} {and}\mspace{14mu} j} = 0}}};$${c\left( {i,j} \right)} = {{\frac{1}{4\sqrt{2}}\mspace{14mu} {if}\mspace{14mu} i} = {{0\mspace{14mu} {and}\mspace{14mu} j} > 0}}$${{{{{or}\mspace{14mu} i} > {0\mspace{14mu} {and}\mspace{14mu} j}} = 0};{{c\left( {i,j} \right)} = {\frac{1}{4}\mspace{14mu} {when}\mspace{14mu} i}}},{j > 0}$

As previously stated, digital images are often transmitted and stored incompressed data formats, such as the previously described JPEG standard.In this context, there often arises the need to scale down (i. e.,reduce) the dimensions of an image that is provided in a compressed dataformat in order to achieve a suitable image size.

For example, where an image is to be sent in compressed data format toreceivers of different computational and output capabilities, it may benecessary to scale down the size of the image to match the capabilitiesof each receiver. For example, some printers are designed to receiveimages which are of a certain size, but the printers must have thecapability of scaling down the image size for printing purposes,particularly when the original image was intended for a higherresolution output.

A known method for scaling down an image provided in a compressed dataformat is illustrated in FIG. 1. First, a determination is made as tothe amount of desired image reduction 1/B. (Block 10) If, for example,an image reduction by a factor of 3 is desired (i.e., dividing by 3 ormultiplying by ⅓), then 3 adjacent 8×8 blocks of transformed data values(for a total of 64×3 or 192 values) are retrieved from the image. (Block11) An IDCT is performed on each of the three blocks to transform thepixel or pel data into the real or spatial domain. (Block 12) Once inthe real domain, the data is reduced from 3 adjacent 8×8 data blocksinto a single 8×8 data block by any one of several, known filteringtechniques. (Block 13) Then a FDCT operation is performed on the data ofthe single 8×8 data block to return the data to the DCT domain. (Block14) The process is repeated for all remaining data in the input image.(Block 15)

Thus given a portion of an image in a JPEG/DCT compressed data formatconsisting of four compressed 8×8 blocks of image data, scaling down theimage by a factor of two in each dimension using a previously knownmethod requires: (1) entropy decoding the data which is inone-dimensional vector format and placing the data in 8×8 blocks; (2)de-quantizing the data; (3) performing four 8×8 IDCT operations toinverse transform the transformed blocks of image data; (4) additionalfiltering operations to scale down the blocks of image data into one 8×8block of scaled image data; (5) an 8×8 FDCT operation to re-transformthe block of scaled image data; (6) quantizing the 8×8 block of data;and (7) placing the block of data in one-dimensional vectors and entropyencoding the data for storage or transmission. Given the mathematicalcomplexity of the FDCT and IDCT operations, such a large number ofoperations is computationally time consuming.

What is needed is an efficient method and apparatus that operatesdirectly upon transformed blocks of image data to convert them intotransformed blocks of scaled-down image data.

SUMMARY OF THE PREFERRED EMBODIMENTS

To overcome the limitations in the prior art described above, preferredembodiments disclose a method, system, and data structure for thescaling down of data. At least two blocks of transformed data samplesrepresenting at least two blocks of original data samples are received.One of at least two tables of constants is selected wherein each tableof constants is capable of reducing the number of transformed datasamples by a different factor. The constants taken from the selectedtable are applied to the at least two blocks of transformed data samplesto produce one block of transformed data samples representing one blockof final data samples.

In another embodiment, a method for generating a plurality of constantsfor use in decreasing the number of original data samples by a factor ofB is provided. An inverse transform operation is applied on B sets oforiginal transform coefficients to obtain B sets of variables in thereal domain as a function of the B sets of original transformcoefficients. The B sets of variables in the real domain are scaled downto produce a single set of variables representing scaled data samples inthe real domain. A forward transform operation is applied to the singleset of variables to obtain a single set of new transform coefficients asa function of the B sets of original transform coefficients. This, inturn, yields the plurality of constants.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates the logic for a known method of scaling down the sizeof an input image which is received in transformed format.

FIG. 2 is a simplified block diagram of a DCT-based JPEG encoder.

FIG. 3 is a simplified block diagram of a DCT-based JPEG decoder.

FIG. 4 is a block diagram of a simple printing system that uses JPEGcompressed images.

FIG. 5 is a simplified block diagram showing the scaling down of thesize of an input image.

FIGS. 6 a and 6 b are simplified block diagrams showing the scaling downof the size of an input image by incorporating the principles of certainembodiments of the present invention.

FIGS. 7 a and 7 b are graphical representations of the scaling down of aportion of a transformed image in accordance with the inventionsdisclosed herein.

FIG. 8 is a graphical representation of the relationship between the DCTcoefficients in a portion of an input transformed image and the DCTcoefficients of a portion of a resulting scaled-down image.

FIG. 9 illustrates the logic to generate a table of constants for thescaling down of data by a factor of B.

FIG. 10 illustrates the logic to scale down the dimensions of the wholeof an input transformed image.

FIG. 11 is a table of values for the selection of tables of constantsfor the scaling up or scaling down of data by integer and non-integeramounts.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the following description, reference is made to the accompanyingdrawings which form a part hereof and which illustrate severalembodiments of the present invention. It is understood that otherembodiments may be used and structural and operational changes may bemade without departing from the scope of the present invention.

For purposes of illustrating the invention, the well-known JPEG and MPEGDCT transform operation of images is being used. However, the sametechniques can be used to compress any two-dimensional array of data.DCT transform operations work best when the data contains some internalcorrelation that the FDCT can then de-correlate.

Computing Environment

Referring now to the drawings, and more particularly to FIG. 2, there isshown a simplified block diagram of a DCT-based encoder. A source imagesampled data 16 in 8×8 blocks are input to the encoder 17. Each 8×8 datablock is transformed by the Forward Discrete Cosine Transform (FDCT) 121into a set of 64 values, referred to as DCT coefficients or transformeddata samples. One of these values is referred to as the DC coefficient,and the other 63 values are referred to as AC coefficients. Each of the64 transformed data samples are then quantized by quantizer 122 usingone of 64 corresponding input quantization values from a quantizationtable 123. The quantized transformed data samples are then passed to anentropy encoding procedure 124 using table specifications 125. Thisprocedure compresses the data further. One of two entropy encodingprocedures can be used, Huffman encoding or arithmetic encoding. IfHuffman encoding is used, the Huffman table specifications must beprovided, but if arithmetic encoding is used, then arithmetic codingconditioning table specifications must be provided. The previousquantized DC coefficient is used to predict the current DC coefficientand the difference is encoded. The 63 AC coefficients, however, are notdifferentially encoded but, rather, are converted into a zig-zagsequence. The output of the entropy encoder is the compressed image data18.

FIG. 3 shows a simplified block diagram of the DCT-based decoder. Eachstep shown performs essentially the inverse of its corresponding mainprocedure within the encoder shown in FIG. 2. The compressed image data18 is input to the decoder 22 where it is first processed by an entropydecoder procedure 221 which decodes the zig-zag sequence of thequantized DCT coefficients. This is done using either Huffman tablespecifications or arithmetic coding conditioning table specifications222, depending on the coding used in the encoder. The quantized DCTcoefficients (or quantized transformed data samples) output from theentropy decoder are input to the dequantizer 223 which, usingquantization table specifications 224 comprising output quantizationvalues, outputs dequantized DCT coefficients to Inverse Discrete CosineTransform (IDCT) 225. The output of the IDCT 225 is the reconstructedimage 20.

FIG. 4 shows a block diagram of a simple printing system that uses JPEGcompressed images. For this illustration, the input image is assumed tobe a grayscale image comprised of samples from only one component andthe printer, a grayscale printer. The input image 41 is scanned in thescanner 42 and then the source image data, a single component of gray,is compressed with a JPEG encoder 43, such as described with referenceto FIG. 2. The JPEG encoder 43 is shown separately from the scanner 42,but in a practical embodiment, the JPEG encoder 43 could be incorporatedwith the scanner 42. The output of the JPEG encoder 43 is compresseddata 44. After optional transmission, the compressed data is stored on adisk storage device 45. At some later time, the compressed data storedon the disk storage device 45 is retrieved by the printer server 46which scales the image in the transformed domain. The scaled image isrecompressed in the printer server 46 so that the JPEG decoder 47decodes the reconstructed scaled-down image. The JPEG decoder 47 is asdescribed with reference to FIG. 3. The printer 48 prints the grayscalescaled image and produces the output image 49 on paper.

Reduction of Image Size

The preferred embodiments of the present invention include a method,system and data structure for efficiently scaling down data which isreceived in a transformed or a DCT-based data format. A one-dimensionalDCT-domain reduction method is disclosed that scales down B blocks ofdata along one dimension into 1 block where B is the reduction factorfor the entire set of data along one axis.

FIG. 5 shows a known, traditional method for the reduction of images.The compressed image 51 is JPEG decoded by JPEG decoder 52. The decoder52 first entropy decodes 53 the data which arrives in one-dimensionalvector format and places the data in two-dimensional block format. Nextthe data is taken through a de-quantizing step 54. Finally, an IDCToperation 55 is performed to inverse transform the transformed blocks ofimage data to the real domain.

When in the real domain, the image data is manipulated 56 by additionalfiltering operations to scale down the blocks of image data into fewerblocks of scaled image data.

The scaled data is sent to the JPEG encoder 57 where the process isreversed. First, FDCT operations 58 are performed to re-transform thereduced number of blocks of scaled image data from the real domain tothe DCT domain. The transformed data is then quantized 59, placed in aone-dimensional vector and entropy encoded 60 for storage ortransmission as a JPEG encoded image of scaled down dimensions 61. Thusit is seen that the known, scale down method of FIG. 5 involvesperforming scale down operations on data which is in the real domain.

FIGS. 6 a and 6 b show the application of the present inventionresulting in a faster scaling down of data. Referring first to FIG. 6 a,the compressed image 71 is JPEG entropy decoded 72 and then de-quantized73. As will be explained in more detail below, the transformed,dequantized data is then manipulated to create fewer blocks oftransformed data while remaining in the DCT domain 74 in contrast tobeing processed in the real domain as shown in FIG. 5. Then the DCTcoefficients of the scaled-down image are re-quantized 75 and JPEGentropy encoded 76 to produce a JPEG encoded image of scaled downdimensions 77. This can greatly increase the speed of processing imagedata in, for example, high speed printers.

FIG. 6 b shows an alternative embodiment of the present invention whichprovides a possibly faster method of scaling down data. The compressedimage 78 is JPEG entropy decoded 79 to produce quantized DCT, ortransformed, data. As will be explained in more detail below, thequantized DCT coefficients are then directly manipulated and scaled downwhile remaining in the DCT domain 80. Then the quantized DCTcoefficients of the scaled down image are entropy encoded 81 to producea JPEG encoded image of scaled down dimensions 82. Thus in contrast tothe embodiment of FIG. 6 a, the embodiment of FIG. 6 b combines thede-quantizing and quantizing steps into the manipulation step 80.

Embodiments of the present invention involve scaling down transformeddata while remaining in the DCT domain by multiplying the DCTcoefficients directly by one or more tables of constants. The result ofsuch multiplication is a reduced number of DCT coefficients, which ifinverse transformed into the real domain, would correspond to a scaleddown image. As can be appreciated by a comparison of the known method ofFIG. 5 with the inventive embodiments of FIGS. 6 a and 6 b, thisrelatively simple multiplication step replaces the more computationallyintensive steps associated with the FDCT and IDCT operations.

As explained in more detail below, a table of constants for use inscaling down a number of original data samples by a factor of B isdeveloped by applying an inverse transform operation on B sets oforiginal transform coefficients to obtain B sets of variables in thereal domain as a function of the B sets of original transformcoefficients. The B sets of variables in the real domain are scaled downto produce a single set of variables representing scaled data samples inthe real domain. A forward transform operation is applied to the singleset of variables to obtain a single set of new transform coefficients asa function of the B sets of original transform coefficients. This, inturn, yields the plurality of constants which can be stored and recalledwhenever desired to directly operate on the input transformed data andderive reduced-sized image transform data.

Referring to FIG. 7 a, a portion of a transformed image is representedby two adjacent 8×8 blocks 201 of DCT coefficients: {tilde over(G)}_(0,0) . . . {tilde over (G)}_(7,7) and {tilde over (H)}_(0,0) . . .{tilde over (H)}_(7,7). If for example a reduction of ½ in the X axis isdesired, the preferred embodiments operate directly upon these DCTcoefficients to produce one 8×8 block 202 of DCT coefficients, {tildeover (F)}_(0,0) . . . {tilde over (F)}_(7,7), which, if inversetransformed, would yield an image of ½ size of the original along the Xaxis.

FIG. 7 a represents the result in one dimension when all data in twoblocks have been processed. In actual practice however, the inventivemethods operate in one dimension on a sub-block 203 consisting of onerow of data as shown in FIG. 7 b. The output sub-blocks, or rows, 204are sequentially created until two complete, original 8×8 data blockshave been scaled down.

FIG. 8 illustrates the relationship between the DCT coefficients in thereduced image and those of the original in an example where an imagereduction amount of ½ in one dimension is desired according to oneembodiment of the present invention. A row of coefficients from twoadjacent 8×8 blocks is retrieved. These rows are represented by {tildeover (G)}_(n,0) . . . {tilde over (G)}_(n,7) and {tilde over (H)}_(n,0). . . {tilde over (H)}_(n,7). Using equation (3), an IDCT operation isperformed on each of these rows which transforms the data from the DCTdomain to the spatial or real domain, giving pixel values, G_(n,0) . . .G_(n,7) and H_(n,0) . . . H_(n,7). To accomplish an image reduction of ½in one dimension in the real domain, each pair of adjacent pixel valuesare averaged. This is a filtering process, and this particular method isreferred to as a low pass filter. This averaged value is represented inFIG. 8 by F_(x,y) where F_(n,0)=½ (G_(n,0)+G_(n,1)), F_(n,1)=½(G_(n,2)+G_(n,3)), . . . F_(n,7)=½ (H_(n,6)+H_(n,7)). Finally, theseaveraged, spatial domain values, F_(x,y), are transformed into the DCTdomain by use of equation (1).

FIGS. 7 a, 7 b and 8 illustrate graphically the relationship between theinput DCT image data, {tilde over (G)}, {tilde over (H)} and the reducedimage DCT data, {tilde over (F)}. Mathematically, this relationship isshown as follows.

Again for example using an image reduction factor of 2 in one dimensionon a one-dimensional data block of 1×8 coefficients, a representation ofa plurality of transform coefficients of a set of variables is made.This is in the form of a row of data in one dimension for two adjacentdata blocks, {tilde over (G)}₀, . . . {tilde over (G)}₇, {tilde over(H)}₀, . . . {tilde over (H)}₇. The relationship between each value inthe spatial domain and its counterpart in the DCT domain is representedby the IDCT equations as follows:

$\begin{matrix}{G_{x} = {\sum\limits_{u = 0}^{7}{C_{u}{\overset{\sim}{G}}_{u}{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; u}{16} \right)}}}} & (5) \\{{H_{x} = {\sum\limits_{u = 0}^{7}{C_{u}{\overset{\sim}{H}}_{u}{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; u}{16} \right)}}}}{where}{C = {{\frac{1}{\sqrt{8}}\mspace{14mu} {for}\mspace{14mu} u} = {{0\mspace{14mu} {and}\mspace{14mu} C} = {{\frac{1}{2}\mspace{14mu} {for}\mspace{14mu} u} > 0}}}}} & (6)\end{matrix}$

Using for example a low pass filtering technique, each pair of thesevalues is averaged to achieve the image reduction of ½ in one dimensionwhich is represented as follows:

F ₀≡½(G ₀ +G ₁),F ₁≡½(G ₂ +G ₃), . . . , F ₇≡½(H ₆ +H ₇)  (7)

The FDCT equation is applied to each spatial domain value of the reducedimage to transform these back into the DCT domain:

$\begin{matrix}{{{\overset{\sim}{F}}_{v} = {C_{v}{\sum\limits_{x = 0}^{7}{F_{x}{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; v}{16} \right)}}}}},{v = 0},\ldots \mspace{11mu},7} & (8)\end{matrix}$

Now, since F₀½ (G₀+G₁), F₁=½ (G₂+G₃), . . . F₇=½ (H₆+H₇), each of theseexpressions for F₀, . . . F₇ can be substituted in equation (8) whichcan then be represented as follows:

$\begin{matrix}{{\overset{\sim}{F}}_{v} = {{C_{v}{\sum\limits_{x = 0}^{3}\; {\left\{ {\frac{1}{2}\left( {G_{2\; x} + G_{{2x} + 1}} \right)} \right\} {\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; v}{16} \right)}}}} + {C_{v\;}{\sum\limits_{x = 4}^{7}\; {\left\{ {\frac{1}{2}\left( {H_{{2x} - 8} + H_{{2x} - 7}} \right)} \right\} {\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; v}{16} \right)}}}}}} & (9)\end{matrix}$

The expressions for G and H from equations (5) and (6) can besubstituted for each occurrence of G and H in equation (9) as follows:

$\begin{matrix}{{\overset{\sim}{F}}_{v} = {C_{v}\left\lbrack {{\sum\limits_{x = 0}^{3}\; {\frac{1}{2} \left\{ {{\sum\limits_{u = 0}^{7}\; {C_{u}{\overset{\sim}{G}}_{u}{\cos \left( \frac{\left( {{4x} + 1} \right)\pi \; u}{16} \right)}}} + {\sum\limits_{u = 0}^{7}\; {C_{u}{\overset{\sim}{G}}_{u}{\cos \left( \frac{\left( {{4x} + 3} \right)\pi \; u}{16} \right)}}}} \right\}  {\cos\left( \frac{\left( {{2x} + 1} \right)\pi \; v}{16} \right)}}} + \left. \quad {\sum\limits_{x = 4}^{7}\; {{\frac{1}{2}\left\lbrack {{\sum\limits_{u = 0}^{7}\; {C_{u}{\overset{\sim}{H}}_{u}{\cos \left( \frac{\left( {{4x} - 15} \right)\pi \; u}{16} \right)}}} + {\sum\limits_{u = 0}^{7}\; {C_{u}{\overset{\sim}{H}}_{u}{\cos \left( \frac{\left( {{4x} - 13} \right)\pi \; u}{16} \right)}}}} \right\}}{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; v}{16} \right)}}} \right\rbrack} \right.}} & (10)\end{matrix}$

From equation (10) it can be appreciated that the values {tilde over(F)} which are reduced image coefficients in the DCT domain now are afunction of {tilde over (G)} and {tilde over (H)} which are the originalsized image coefficients, also in the DCT domain. Equation (10) can bealgebraically simplified into the following representation:

$\begin{matrix}{{\overset{\sim}{F}}_{v} = {\frac{1}{2}{\sum\limits_{u = 0}^{7}\; {C_{u}C_{v}\left\{ \left\lbrack {{\sum\limits_{x = 0}^{3}\; {\cos \left( \frac{\left( {{4x} + 1} \right)\pi \; u}{16} \right)}} + {\left. \quad{\cos\left( \frac{\left( {{4x} + 3} \right)\pi \; u}{16} \right)} \right\rbrack  {\cos\left( \frac{\left( {{2\; x} + 1} \right)\pi \; v}{16} \right)} {\overset{\sim}{G}}_{u}} + {\quad {{{\left\lbrack {{\underset{x = 4}{\overset{7}{\sum}}\; {\cos\left( \frac{\left( {{4x} - 15} \right)\pi \; u}{16} \right)}} + {\left. \quad{\cos \left( \frac{\left( {{4x} - 13} \right)\pi \; u}{16} \right)} \right\rbrack {\cos \left( \frac{\left( {{2\; x} + 1} \right)\pi \; v}{16} \right)}{\overset{\sim}{H}}_{u}}} \right\} \mspace{20mu} {for}{\mspace{14mu} \;}v} = 0},\ldots \mspace{14mu},{7\mspace{20mu} {where}\mspace{14mu} C_{u}},{C_{v} = {\frac{1}{\sqrt{8}}\mspace{14mu} {for}\mspace{14mu} u}},{v = {0\mspace{14mu} {and}\mspace{14mu} \mspace{20mu} C_{u}}},{C_{v} = {\frac{1}{2}\mspace{14mu} {for}\mspace{14mu} u}},{v > 0}}}} \right. \right.}}}} & (11)\end{matrix}$

From equation (11) it is apparent that not only is {tilde over (F)} afunction of {tilde over (G)} and {tilde over (H)}, but that thisrelationship involves only constants. Appendix A contains an exemplarytable of constants which are used to obtain a ½ reduction in a onedimensional row for values {tilde over (F)}₀, . . . {tilde over (F)}₇,using a low pass filtering technique.

Thus for example, taking the first row of values from the {tilde over(G)} block matrix and the first row from the {tilde over (H)} blockmatrix of Appendix A,

$\begin{matrix}{{\overset{\sim}{F}}_{0} = {{0.50000{\overset{\sim}{G}}_{0}} - {0.00000{\overset{\sim}{G}}_{1}} + {0.00000{\overset{\sim}{G}}_{2}} - {0.00000{\overset{\sim}{G}}_{3}} +}} \\{{{0.00000{\overset{\sim}{G}}_{4}} - {0.00000{\overset{\sim}{G}}_{5}} - {0.00000{\overset{\sim}{G}}_{6}} - {0.00000{\overset{\sim}{G}}_{7}} +}} \\{{{0.50000{\overset{\sim}{H}}_{0}} - {0.00000{\overset{\sim}{H}}_{1}} + {0.00000{\overset{\sim}{H}}_{2}} - {0.00000{\overset{\sim}{H}}_{3}} +}} \\{{{0.00000{\overset{\sim}{H}}_{4}} - {0.00000{\overset{\sim}{H}}_{5}} + {0.00000{\overset{\sim}{H}}_{6}} -}} \\{{0.00000{\overset{\sim}{H}}_{7}} = {{0.50000{\overset{\sim}{G}}_{0}} + {0.50000{\overset{\sim}{H}}_{0}}}}\end{matrix}$

Similarly for example, taking the second row of values from the {tildeover (G)} block matrix and the second row of values from the {tilde over(H)} block matrix of Appendix A,

${\overset{\sim}{F}}_{1} = {{0.45306{\overset{\sim}{G}}_{0}} + {0.20387{\overset{\sim}{G}}_{1}} - {0.03449{\overset{\sim}{G}}_{2}} + {0.00952{\overset{\sim}{G}}_{3}} + {0.00000{\overset{\sim}{G}}_{4}} - {0.00636{\overset{\sim}{G}}_{5}} + {0.01429{\overset{\sim}{G}}_{6}} - {0.04055{\overset{\sim}{G}}_{7}} - {0.45306{\overset{\sim}{H}}_{0}} + {0.20387{\overset{\sim}{H}}_{1}} + {0.03449{\overset{\sim}{H}}_{2}} + {0.00952{\overset{\sim}{H}}_{3}} - {0.00000{\overset{\sim}{H}}_{4}} - {0.00636{\overset{\sim}{H}}_{5}} - {0.01429{\overset{\sim}{H}}_{6}} - {0.04055{\overset{\sim}{H}}_{7}}}$

Equation (11) yields a table of constants based upon a low pass filtertechnique where F₀=½(G₀+G₁), F₁=½(G₂+G₃), etc. However, a similarmethodology as that described above can be used to obtain alternativeequations and corresponding tables of constants which are based uponother types of filters.

For example instead of a low pass filter which is based upon theaveraging of two values, a 1:2:1 filter, collocated on the left, couldbe used. Data filtered by this algorithm is represented asF₀=(3G₀+G₁)/4, F₁=(G₁+2G₂+G₃)/4, F₂=(G₃+2G₄+G₅)/4, . . . ,F₇=(H₅+2H₆+H₇)/4. Similarly, a 1:2:1 filter, collocated on the right,could be used. This is represented by F₀=(G₁+2G₂+G₃)/4,F₁=(G₃+2G₄+G₅)/4, . . . , F₆=(H₄+2H₅+H₆)/4, F₇=(H₆+3H₇)/4. Other filterscan be employed without departing from the spirit of the invention.

FIG. 9 illustrates the logic for a generalized approach in developing atable of constants in accordance with the previously describedembodiment. At block 501 a determination is made as to the amount ofimage reduction, 1/B, where B is an integer. Thus for example if animage size reduction of ⅓ is desired, then B is set equal to 3. At block502, a representation of a set of image data is created in the form of Bequations for one row of B adjacent blocks of data. In the instancewhere B=3, the 3 exemplary equations could be {tilde over (G)}=({tildeover (G)}₀, . . . , {tilde over (G)}₇), {tilde over (H)}=({tilde over(H)}₀, . . . , {tilde over (H)}₇), and Ĩ=(Ĩ₀, . . . , Ĩ₇).

Next an IDCT is performed on the row of B adjacent data blocks in orderto obtain real domain expressions. (Block 503). The real domainrepresentations for B blocks are reduced to 1 block by a filteringtechnique of choice, whereupon a FDCT is performed on the 1 block ofreal domain expressions. (Blocks 504 & 505). These equations are nextsolved so that the new data block's coefficients are expressed as afunction of the original B blocks' coefficients. (Block 506) It has beenshown that this relationship results in a group of constants which canthen be stored in a Table B for future use in reducing any input imagein DCT format by 1/B of the original size in one dimension. (Block 507)

The method of FIG. 9 can be repeated as many times as desired fordifferent reduction amounts, such as ½, ⅓, ¼, ⅕, etc. Different tablesof constants can be generated for each reduction amount and can bestored in computer memory. Thus a computer system containing a set ofsuch tables will be able to rapidly perform an image reduction of anyamount selected by the user which corresponds to a table of constants.The constants from these tables can be applied directly by multiplyingthem by the input image DCT coefficients. This results in an outputimage of reduced size in the DCT domain.

An alternative technique for developing a representation of the DCTcoefficients of a reduced image as a function of an original image'scoefficients is as follows. Again, taking the example of reducing theoriginal image by ½ in one dimension, a set of data representing a rowfrom two adjacent blocks, G and H, in the real domain are taken andrepresented to be equal to one larger block F, where F=(F₀, . . . ,F₁₅), as follows:

$\begin{matrix}{{{F_{0} = G_{0}},{F_{1} = G_{1}},\ldots \mspace{20mu},{F_{8} = H_{0}},\ldots \mspace{14mu},{F_{15} = H_{7}},{or}}{F_{u} \equiv {{\sum\limits_{u = 0}^{7}\; G_{u}} + {\sum\limits_{u = 8}^{15}\; H_{({u - 8})}}}}} & (12)\end{matrix}$

Using a scaled version of the FDCT equation (1), the transformcoefficients for F (a block of 16 values) can be represented as:

$\begin{matrix}{{{{\overset{\sim}{F}}_{u} = {C_{u}{\sum\limits_{x = 0}^{15}\; {F_{x}{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; u}{32} \right)}}}}},{u = 0},\ldots \mspace{14mu},7,\left( {{{{or}\mspace{14mu} {alternatively}\mspace{14mu} u} = 0},\ldots \mspace{14mu},15} \right)}{where}{C_{u} = {{\frac{1}{2\sqrt{2}}\mspace{14mu} {for}\mspace{14mu} u} = {{0\mspace{14mu} {and}\mspace{14mu} C_{u}} = {{\frac{1}{2}\mspace{14mu} {for}\mspace{14mu} u} > 0}}}}} & (13)\end{matrix}$

Note that the C_(u) is defined as an 8 sample DCT notwithstanding that a16 sample DCT is in fact being used for equation 13. Hence these arereferred to as “scaled” DCT transforms. Note also that while equation(13) is valid for all values of u=0, . . . , 15, this is not necessaryin most cases. Since we normally are only concerned with the lowesteight frequencies, this equation can be simplified by discarding theupper eight frequencies and using only u=0, . . . 7.

Since G_(x)=F_(x) and H_(x)=F_(x+8) for x=0, . . . ,7, we can substitutethese in equation (13) as follows:

$\begin{matrix}\begin{matrix}{{\overset{\sim}{F}}_{u} = {C_{u}\left( {{\sum\limits_{x = 0}^{7}\; {G_{x}{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; u}{32\;} \right)}}} +} \right.}} \\\left. {\sum\limits_{x = 8}^{15}\; {H_{({x - 8})}{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; u}{32} \right)}}} \right) \\{{{{for}\mspace{14mu} u} = 0},\ldots \mspace{14mu},7}\end{matrix} & (14)\end{matrix}$

Using the scaled version of the IDCT equation (3), we have therelationships:

$\begin{matrix}{G_{x} = {\sum\limits_{v = 0}^{7}\; {C_{v\;}{\overset{\sim}{G}}_{v}{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; v}{16} \right)}}}} & (15) \\{{{H_{x} = {\sum\limits_{v = 0}^{7}\; {C_{v}{\overset{\sim}{H}}_{v}{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; v}{16} \right)}}}},{x = 0},\ldots \mspace{14mu},7}{where}{C_{v} = {{\frac{1}{2\sqrt{2}}\mspace{14mu} {for}\mspace{14mu} v} = {{0\mspace{14mu} {and}{\mspace{11mu} \;}C_{v}} = {{\frac{1}{2}\mspace{14mu} {for}\mspace{14mu} v} > 0}}}}} & (16)\end{matrix}$

Substituting these values for G_(x) and H_(x) into equation (14) yieldsthe final representation:

$\begin{matrix}{{\overset{\sim}{F}}_{u} = {\sum\limits_{v = 0}^{7}\; {C_{u}C_{v}\left\{ {{{{\left\lbrack {\sum\limits_{x = 0}^{7}\; {{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; v}{16} \right)}{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; u}{32} \right)}}} \right\rbrack  {\overset{\sim}{G}}_{v}} + {\left. \quad{\left\lbrack {\sum\limits_{x = 0}^{7}\; {{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; v}{16} \right)}{\cos \left( \frac{\left( {{2x} + 17} \right)\pi \; u}{32} \right)}}} \right\rbrack {\overset{\sim}{H}}_{v}} \right\} \mspace{20mu} {for}\mspace{14mu} u}} = 0},\ldots \mspace{14mu},{7\mspace{20mu} {where}\mspace{20mu} C_{u}},{C_{v} = {\frac{1}{2\sqrt{2}}\mspace{14mu} {for}\mspace{14mu} u}},{v = {0\mspace{14mu} {and}\mspace{14mu} \mspace{20mu} C_{u}}},{C_{v} = {\frac{1}{2}\mspace{11mu} {for}\mspace{14mu} u}},{v > 0}} \right.}}} & (17)\end{matrix}$

As before, it is apparent from equation (17) that not only is {tildeover (F)} a function of {tilde over (G)} and {tilde over (H)}, but thatthis relationship involves only constants. Appendix B contains anexemplary table of constants which are used to obtain a ½ reduction in aone dimensional row for values {tilde over (F)}₀, . . . {tilde over(F)}₇. It will be noted that the constants in both Appendices A and B,although different, can be used to achieve a ½ image reduction. Thechoice is left to the user based upon user needs and preferences.

Once again, it can be appreciated that although the above describedembodiment is illustrative of a reduction of ½ in one dimension, similarmethodologies can be employed to develop equations and correspondingtables of constants for other reduction amounts 1/B, where B=3, 4, 5,etc.

FIG. 10 illustrates the steps for employing these tables of constants toreduce the size of an input image. The process commences with thereceipt of an image file in DCT format. (Blocks 601 & 602). Adetermination is made of the amount of image scale down in both theX-axis and Y-axis. (Block 603) The reduction amounts do not have to beequal in both axes. Thus for example if the user desired a ½ image scaledown for the X-axis and a ¼ scale down for the Y-axis, then for purposesof FIG. 10, S=2 and T=4.

Next a scale down procedure along the X-axis is commenced by retrievingS adjacent blocks of DCT coefficients along the X-axis. (Block 604) The1^(st) row or sub-block of S×8 coefficients from the adjacent blocks isretrieved (block 606) and used to calculate the 1^(st) row of 1×8reduced image DCT coefficients using the constants previously stored inTable S. Thus 8 coefficients would be calculated. (Block 607) Thesecoefficients would represent the 1^(st) row of a single 1×8 data blockalong the X-axis.

Table S is one of two or more tables of constants stored in computermemory, each of which can be used to scale down image data by adifferent factor. In this example, Table S contains the constants whichcorrespond to a scale down factor of 2. This calculation is accomplishedby multiplying the S×8, or 16, coefficients of the original image by theconstants from Table S.

The process of blocks 606 and 607 is repeated for rows or sub-blocks 2through 8 of the retrieved, adjacent S data blocks to thereby generateone 8×8 block of reduced image coefficients along the X-axis. Next, adetermination is made whether the original image contains additionaldata blocks along the X-axis to be processed. (Block 609) If so, controlloops to block 604 where the next group of S blocks of DCT coefficientsare retrieved. The previously described process continues until allX-axis image data has been scaled down.

When there is no longer any X-axis image data to scale down, controltransfers to block 610 where T adjacent blocks of DCT coefficients alongthe Y-axis are retrieved. In the case where a Y-axis image scale downfactor of 3 is selected, then 3 adjacent data blocks would be retrieved.The process of blocks 610 through 615 of FIG. 10 is the same as thatdescribed for blocks 604 through 609 except that now the calculationsproceed along columns of data for Y-axis reduction, rather than alongrows of data.

From FIGS. 6 a and 10 it can be seen that by maintaining tables ofconstants in memory for various scale down factors, such as ½, ⅓, ¼, ⅕,etc., it is possible to rapidly transform a larger image which isrepresented by a file containing DCT domain data into a reduced sizedimage by multiplying the DCT domain data with these constants. It is notnecessary to perform an IDCT to transform the image into the spatialdomain, reduce the image size while in the spatial domain, and thenperform a FDCT to again transform the image file—a procedure which iscomputationally much more intensive.

The foregoing has been described with respect to data blocks of 8×8data. However it should be appreciated that the inventions claimedherein can apply to any data block which, for these purposes, is meantto be any set of data including, but not limited to, one or twodimensional arrays of data of any size.

Referring again to FIG. 6 b, it further should be appreciated by thoseskilled in the art that an alternative embodiment of the presentinvention involves the incorporation of the de-quantization andre-quantization steps into the tables of constants, or matrices, in thefollowing way:

Assume that the constants defining a particular block of a particularone dimensional transform-based scale-down scheme are known to beD=(d_(ij)) with i,j=0, . . . , 7. Let Q=(q_(i)) and R=(r_(i)) (with i=0,. . . , 7) be the desired quantization vectors for the input and output,respectively. These quantizations can be incorporated into the transformmatrix D to obtain new constants C=(c_(ij)) wherec_(ij)=d_(ij)·(q_(j)/r_(i)). Thus by including the de-quantization andre-quantization values in the tables of constants, it can be seen thateven faster scaling down operations can be realized.

Thus using this method, a plurality of input and output quantizationvalues is received. Also a block of quantized transformed data samplesassociated with the input quantization values is received wherein theblock of quantized transformed data samples represents a block oforiginal data samples. A table of constants capable of decreasing thenumber of transformed data samples by a different factor is selected.The plurality of input and output quantization values are applied to theselected table of constants to produce a plurality of new constants. Theplurality of new constants is applied to the blocks of quantizedtransformed data samples to produce one block of quantized transformeddata samples associated with the output quantization values wherein thequantized transformed data samples represent one block of final datasamples.

An advantage of the approaches of either FIG. 6 a or FIG. 6 b is that itis not necessary to handle 64×N samples during the reduction process. Itis a characteristic of the JPEG standard that the original, quantizedDCT coefficients are usually rather sparse and that most rows in theblocks contain coefficients which are a value of zero. Thus under thismethod, the reduced blocks will keep these rows as zeros. Other rowsfrequently have only 1, 2 or 3 coefficients that are non-zero. Thesecases can be handled with much simplified equations that have discardedthe zero terms.

Another advantage of the disclosed embodiments relates to the conductingof these computations in one dimension. Blocks of data are scaled downindependently along each axis. Thus one is not restricted to identicalscaling in the X and Y directions. For example, an image could bereduced by a factor 3 on the X-axis and by a factor of 2 on the Y-axis.

Because the disclosed method incorporates independent treatment of eachaxis in one dimension, it is further possible to scale up the image sizeon one axis and reduce the size on the other axis. In an alternativeembodiment, the scaling up of images in DCT format can be accomplishedthrough the use of tables of constants which are derived from equationsbased upon scaling up algorithms. These equations and constants arederived as follows.

Using for example a scale up factor of 2 in one dimension on a 1×8 pixelor pel row of values from a data block in the real domain, arepresentation of a set of image data is made. This is in the form of arow of data, F₀, . . . F₇. Since in this example the image is to beenlarged or scaled up by a factor of two in one dimension, the valuesfor F₀, . . . F₇ are mathematically interpolated to form an enlarged1×16 block of pixel or pel values, F₀, ½(F₀+F₁), F₁, ½(F₁+F₂), F₂, . . ., F₇. Next, this 1×16 block of values is re-defined to be the same astwo, adjacent 1×8 blocks, G₀, . . . , G₇ and H₀, . . . , H₇.

Mathematically, this can be represented as follows:

$\begin{matrix}{{G = \left( {F_{0},{\frac{1}{2}\left( {F_{0} + F_{1}} \right)},F_{1},{\frac{1}{2}\left( {F_{1} + F_{2}} \right)},F_{2},{\frac{1}{2}\left( {F_{2} + F_{3}} \right)},F_{3},{\frac{1}{2}\left( {F_{3} + F_{4}} \right)}} \right)}{H = \left( {{\frac{1}{2}\left( {F_{3} + F_{4}} \right)},F_{4},{\frac{1}{2}\left( {F_{4} + F_{5}} \right)},F_{5},{\frac{1}{2}\left( {F_{5} + F_{6}} \right)},F_{6},{\frac{1}{2}\left( {F_{6} + F_{7}} \right)},F_{7}} \right)}} & (18)\end{matrix}$

Note that with this method, the two interpolated pel values between F₃and F₄ are equal to one another.

Next, G and H are represented in the DCT domain by use of the FDCTequation (1) as follows:

$\begin{matrix}{{{{{\overset{\sim}{G}}_{u} = {{C_{u}{\sum\limits_{x = 0}^{7}\; {G_{x}{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; u}{16} \right)}\mspace{14mu} {for}\mspace{14mu} u}}} = 0}},\ldots \mspace{14mu},7}{and}{{{\overset{\sim}{H}}_{u} = {{C_{u}{\sum\limits_{x = 0}^{7}\; {H_{x}{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; u}{16} \right)}\mspace{14mu} {for}\mspace{14mu} u}}} = 0}},\ldots \mspace{14mu},7}{where}{{C_{u} = {{\frac{1}{2\sqrt{2}}\mspace{14mu} {for}\mspace{14mu} u} = 0}},{C_{u} = {{\frac{1}{2}\mspace{14mu} {for}\mspace{14mu} u} > 0}}}{{and}{\mspace{11mu} \;}{where}}{G_{x} = \begin{Bmatrix}{F_{x/2},{x = {0,2,4,6}}} \\{{\frac{1}{2}\left( {F_{{({x - 1})}/2} + F_{{({x + 1})}/2}} \right)},{x = {1,3,5,7}}}\end{Bmatrix}}{and}{H_{x} = \begin{Bmatrix}{{\frac{1}{2}\left( {F_{{({x + 6})}/2} + F_{{({x + 8})}/2}} \right)x} = {0,2,4,6}} \\{F_{{({x + 7})}/2},{x = {1,3,5,7}}}\end{Bmatrix}}}} & (19)\end{matrix}$

Since the G_(x)'s and H_(x)'s in equation (19) can also be expressed interms of the original 8 samples F₀, . . . , F₇, we can use the IDCTrelation of equation (3) to express the DCT coefficients for G and H asgiven in equation (19) in terms of the original DCT coefficients {tildeover (F)}₀. . . {tilde over (F)}₇ for the sample block F:

$\begin{matrix}{{{F_{x} = {{\sum\limits_{u = 0}^{7}\; {C_{u}{\overset{\sim}{F}}_{u}{\cos \left( \frac{\left( {{2x} + 1} \right)\pi \; u}{16} \right)}\mspace{14mu} {for}\mspace{14mu} x}} = 0}},\ldots \mspace{14mu},7}{where}{{C_{u} = {{\frac{1}{2\sqrt{2}}\mspace{14mu} {for}\mspace{14mu} u} = 0}},{C_{u} = {{\frac{1}{2}\mspace{14mu} {for}\mspace{14mu} u} > 0}}}} & (20)\end{matrix}$

By substituting equation (20) into equation (19) and regrouping, it ispossible to obtain the following:

$\begin{matrix}{{{\overset{\sim}{G}}_{v} = {\frac{C_{v}}{2}{\sum\limits_{u = 0}^{7}\; {C_{u}\left\{ {{{K_{1}(v)}{\cos \left( \frac{\pi \; u}{16} \right)}} + {{K_{2}(v)}{\cos \left( \frac{3\; \pi \; u}{16} \right)}} + {{K_{3}(v)}{\cos \left( \frac{5\; \pi \; u}{16} \right)}} + {{K_{4}(v)}{\cos \left( \frac{7\; \pi \; u}{16} \right)}} + {{\cos \left( \frac{15\; \pi \; v}{16} \right)}{\cos \left( \frac{9\; \pi \; u}{16} \right)}}} \right\} {\overset{\sim}{F}}_{u}}}}}{and}} & (21) \\{{\overset{\sim}{H}}_{v} = {\frac{C_{v}}{2} {\sum\limits_{u = 0}^{7}\; {C_{u} \left\{ {{{\cos \left( \frac{\pi \; v}{16} \right)}{\cos \left( \frac{7\; \pi \; u}{16} \right)}} + {{K_{5}(v)}{\cos \left( \frac{9\; \pi \; u}{16} \right)}} + {\left. \quad{{{K_{6}(v)}{\cos \left( \frac{11\; \pi \; u}{16} \right)}} + {{K_{7}(v)}{\cos \left( \frac{13\; \pi \; u}{16} \right)}} + {{K_{8}(v)}{\cos \left( \frac{15\; \pi \; u}{16} \right)}}} \right\}  {\overset{\sim}{F}}_{u}}} \right.}}}} & (22)\end{matrix}$

${{{where}\mspace{14mu} C_{u}{\mspace{11mu} \;}{and}\mspace{14mu} C_{v}} = {\frac{1}{2\sqrt{2}}\mspace{14mu} {for}\mspace{14mu} u}},{v = {0\mspace{14mu} {and}}}$${{{where}\mspace{14mu} C_{u}\mspace{14mu} {and}\mspace{14mu} C_{v}} = {\frac{1}{2}\mspace{14mu} {for}\mspace{14mu} u}},{v > {0\mspace{14mu} {and}}}$where  K₁(v) = 2 D₁^(v) + D₃^(v), K₂(v) = D₃^(v) + 2 D₅^(v) + D₇^(v), K₃(v) = D₇^(v) + 2 D₉^(v) + D₁₁^(v), K₄(v) = D₁₁^(v) + 2 D₁₃^(v) + D₁₅^(v), K₅(v) = D₁^(v) + 2 D₃^(v) + D₅^(v), K₆(v) = D₅^(v) + 2 D₇^(v) + D₉^(v), K₇(v) = D₉^(v) + 2 D₁₁^(v) + D₁₃^(v), K₈(v) = D₁₃^(v) + 2 D₁₅^(v)${{and}\mspace{14mu} D_{x}^{v}} \equiv {\cos \left( \frac{x\; \pi \; v}{16} \right)}$

From equations (21) and (22), it can be appreciated that the values{tilde over (G)} and {tilde over (H)}, which are the enlarged imagecoefficients in the DCT domain, now are expressed as a function of{tilde over (F)} which are the originally sized image coefficients, alsoin the DCT domain. As before with the scale-down algorithms, equations(21) and (22) show that not only are {tilde over (G)} and {tilde over(H)} a function of {tilde over (F)}, but that this relationship involvesonly constants.

Equations (21) and (22) yield a table of constants based upon thepiecewise-linear interpolation technique of equation (18). However, asimilar methodology as that described above can be used to obtainalternative equations and corresponding tables of constants which arebased upon other types of image enlargement techniques. For example,instead of using a piecewise-linear interpolation method, replicationmethods (i.e., repeating each image pel N times per axis) or spline fitmethods (i.e., using polynomial curves) may be employed withoutdeparting from the spirit of the invention.

Moreover, alternative embodiments of scale-up methods and apparatusesare described in greater detail in co-pending application Ser. No.09/570,382, filed concurrently herewith (IBM Docket BLD9-99-0041) whichapplication is incorporated herein by reference in its entirety.

A mixing and matching of tables of constants for scaling up and scalingdown can extend to sequential operations on the same image to achievescaling by a ratio of integer amounts. When applied alone, the tables ofconstants only scale an image by integer amounts. That is, if an imagereduction, or scale down, is desired, then applying only one table ofconstants to an image permits image reduction of 1/B, where B is aninteger. Similarly, applying only one table of constants for imageenlargement, or scaling up, permits enlargement by a factor of A, whereA is an integer.

However, by applying a combination of tables, one each for scaling upand scaling down, to image data, then additional non-integer resizingfactors can be achieved. For example, if it is desired to enlarge animage by a factor of 2.5, such would not be possible by the use of onescale-up table alone. No integer value for A will result in a scale-upfactor of 2.5. However, a table of constants for scaling up an image bya factor of 5 can be used followed by another table for scaling down theenlarged image by a factor of 2. The final image therefore would beresized by a value of 5/2 or 2.5.

Thus it can be seen that by employing any number of combinations oftables for the scaling up and scaling down of an image, a large varietyof resizing factors can be achieved. FIG. 11 is a table of valuesillustrating various integer and non-integer resizing factors which canbe achieved with a set of 22 tables—11 for scaling up data by factors of2 through 12, and 11 tables for scaling down data by ½ through 1/12. InFIG. 11, dots (.) are placed in those locations where the values arerepeated in the table of FIG. 11 and thus are not necessary.

Still referring to FIG. 11, if for example it is desired to create a newimage which is 0.75 of the size of the original image, then that valueis located in the table. It can be seen that 0.75 corresponds to anumerator A of 3 and a denominator B of 4. By taking image data andapplying the table of constants corresponding to a scale up of 3,followed by the application of a second table of constants correspondingto a scale down of ¼, then the resulting image would be resized by afactor of ¾, or 0.75, of the original.

The values in the table of FIG. 11 can be placed in a computer look-uptable format whereby the numerators and denominators corresponding tomultiple, non-integer and integer resizing factors can be easilyretrieved, and the tables of constants corresponding to those numeratorsand denominators can be sequentially employed for any given input,transformed image data.

Although the table of FIG. 11 illustrates resizing factors for A and Bequal to 1 through 12, it will be appreciated that tables similar toFIG. 11, but using a far greater number of values for A and B, can beemployed.

In summary, preferred embodiments disclose a method, system and datastructure for reducing the size of an input image in transformed format.(These could have come from compressed data that has been entropydecoded.) At least two blocks of transformed data samples representingat least two blocks of original data samples are received. One of atleast two tables of constants is selected wherein each table ofconstants is capable of reducing the number of transformed data samplesby a different factor. The constants taken from the selected table areapplied to the at least two blocks of transformed data samples toproduce one block of transformed data samples representing one block offinal data samples.

In another embodiment, a method for generating a plurality of constantsfor use in decreasing a number of original data samples by a factor of Bis provided. An inverse transform operation is applied on B sets oforiginal transform coefficients to obtain B sets of variables in thereal domain as a function of the B sets of original transformcoefficients. The B sets of variables in the real domain are scaled downto produce a single set of variables representing scaled data samples inthe real domain. A forward transform operation is applied to the singleset of variables to obtain a single set of new transform coefficients asa function of the B sets of original transform coefficients. This, inturn, yields the plurality of constants.

The foregoing description of the preferred embodiments of the inventionhas been presented for the purposes of illustration and description. Itis not intended to be exhaustive or to limit the invention to theprecise form disclosed. Many modifications and variations are possiblein light of the above teaching. It is intended that the scope of theinvention be limited not by this detailed description, but rather by theclaims appended hereto. The above specification, examples and dataprovide a complete description of the manufacture and use of thecomposition of the invention. Since many embodiments of the inventioncan be made without departing from the spirit and scope of theinvention, the invention resides in the claims hereinafter appended.

APPENDIX A

Constant matrix defining transform from the {tilde over (G)} block tothe {tilde over (F)} block of a 2 to 1 scale-down.

0.50000 −0.00000 0.00000 −0.00000 0.00000 −0.00000 0.00000 −0.000000.45306 0.20387 −0.03449 0.00952 0.00000 −0.00636 0.01429 −0.04055−0.00000 0.49039 −0.00000 0.00000 −0.00000 0.00000 −0.00000 −0.09755−0.15909 0.38793 0.23710 −0.04055 −0.00000 0.02710 −0.09821 −0.077160.00000 −0.00000 0.46194 −0.00000 0.00000 −0.00000 −0.19134 0.000000.10630 −0.17284 0.35485 0.20387 0.00000 −0.13622 −0.14698 0.03438−0.00000 0.00000 −0.00000 0.41573 −0.00000 −0.27779 0.00000 −0.00000−0.09012 0.13622 −0.17338 0.35991 −0.00000 −0.24048 0.07182 −0.02710

Constant matrix defining transform from the {tilde over (H)}{tilde over( )}block to the {tilde over (F)} block of a 2 to 1 scale-down.

0.50000 −0.00000 0.00000 −0.00000 0.00000 −0.00000 0.00000 −0.00000−0.45306 0.20387 0.03449 0.00952 −0.00000 −0.00636 −0.01429 −0.040550.00000 −0.49039 0.00000 −0.00000 0.00000 −0.00000 0.00000 0.097550.15909 0.38793 −0.23710 −0.04055 −0.00000 0.02710 0.09821 −0.077160.00000 −0.00000 0.46194 −0.00000 0.00000 −0.00000 −0.19134 0.00000−0.10630 −0.17284 −0.35485 0.20387 −0.00000 −0.13622 0.14698 0.034380.00000 −0.00000 0.00000 −0.41573 0.00000 0.27779 −0.00000 0.000000.09012 0.13622 0.17338 0.35991 −0.00000 −0.24049 −0.07182 −0.02710

APPENDIX B

Constant matrix defining transform from the {tilde over (G)} block tothe {tilde over (F)} block of a 2 to 1 scale-down.

0.50000 0.09012 0.00000 0.10630 0.00000 0.15909 0.00000 0.45306 0.453060.28832 0.03733 0.08668 0.09012 0.10913 0.18767 0.38399 −0.00000 0.490390.17678 0.00000 0.19134 0.00000 0.42678 −0.09755 −0.15909 0.340100.38409 0.05735 0.10630 0.17362 0.25664 −0.31765 0.00000 −0.037330.46194 0.25664 0.00000 0.38409 −0.19134 −0.18767 0.10630 −0.182350.25664 0.42362 0.15909 0.19265 −0.38409 −0.01346 −0.00000 0.00000−0.07322 0.41573 0.46194 −0.27779 −0.17678 −0.00000 −0.09012 0.13399−0.18767 0.24443 0.45306 −0.41332 0.03733 −0.03832

Constant matrix defining transform from the {tilde over (H)} block tothe {tilde over (F)} block of a 2 to 1 scale-down.

0.50000 0.09012 0.00000 0.10630 0.00000 0.15909 0.00000 0.45306 −0.453060.11943 0.10630 −0.06765 0.09012 −0.12184 0.15909 −0.46510 0.00000−0.49039 −0.17678 −0.00000 −0.19134 −0.00000 −0.42678 0.09755 0.159090.43577 −0.09012 −0.13846 0.10630 −0.11943 0.45306 0.16332 0.00000−0.03733 0.46194 0.25664 0.00000 0.38409 −0.19134 −0.18767 −0.10630−0.16332 −0.45306 −0.01587 0.15909 −0.46510 −0.09012 0.08221 0.00000−0.00000 0.07322 −0.41573 −0.46194 0.27779 0.17678 0.00000 0.090120.13846 0.15909 0.47540 0.45306 −0.06765 −0.10630 −0.01587

1. A computer readable medium containing a data structure for scaling down a plurality of original data samples by a factor of B, the data structure comprising: a plurality of values, which when executed by a processor, effect the scaling down of the original data samples by applying the plurality of values to transformed data samples representing the plurality of original data samples, the plurality of values being determined by: (a) expressing a first set of variables as B sets of variables, wherein the first set of variables and the B sets of variables represent data samples; and (b) applying transform operations on the first set of variables and the B sets of variables to obtain a transformation from transform coefficients of the B sets of variables into transform coefficients of the first set of variables by: applying a forward transform operation on the first set of variables to create a second representation of the transform coefficients of the first set of variables as a function of the first set of variables; manipulating the second representation to create a third representation of the transform coefficients of the first set of variables as a function of the B sets of variables; applying inverse transform operations on the B sets of variables to create a fourth representation, the fourth representation being the B sets of variables as a function of the transform coefficients of the B sets of variables; and manipulating the third representation by substituting the fourth representation in the third representation to create a final representation, the final representation being the transform coefficients of the first set of variables as a function of the transform coefficients of the B sets of variables.
 2. The computer readable medium of claim 1, wherein the plurality of values are further determined by applying a plurality of quantization values to the plurality of values to produce a plurality of new values, the plurality of new values being capable of decreasing the number of original data samples.
 3. A system for generating a plurality of constants for use in scaling down a number of original data samples by a factor of B, the system comprising: a processing unit capable of executing software routines; and program logic executed by the processing unit, comprising: means for expressing a first set of variables as B sets of variables, wherein the first set of variables and the B sets of variables represent data samples; means for applying transform operations on the first set of variables and the B sets of variables to obtain a transformation from transform coefficients of the B sets of variables into transform coefficients of the first set of variables by: means for applying a forward transform operation on the first set of variables to create a second representation of the transform coefficients of the first set of variables as a function of the first set of variables; means for manipulating the second representation to create a third representation of the transform coefficients of the first set of variables as a function of the B sets of variables; means for applying inverse transform operations on the B sets of variables to create a fourth representation, the fourth representation being the B sets of variables as a function of the transform coefficients of the B sets of variables; and means for manipulating the third representation by substituting the fourth representation in the third representation to create a final representation, the final representation being the transform coefficients of the first set of variables as a function of the transform coefficients of the B sets of variables.
 4. The system of claim 3, further comprising means for applying a plurality of quantization values to the plurality of constants to produce a plurality of new constants, the plurality of new constants being capable of decreasing the number of original data samples. 